I don't believe one exists, but here's the question:
What is the largest $x \in \mathbb{N}$ such that it cannot be represented in any of the following forms $a,b \in \mathbb{N}$...
$6ab+a+b-1$
$6ab-a-b-1$
$6ab+a-b-1$
Either (a) find the largest $x$, (b) prove there is a largest $x$ without finding it, or (c) prove that there is no largest $x$.
After multiplying by 6 and some fiddling around, your representation of $x$ may be reformulated to state that of the following should hold: $$\begin{cases}6x + 7 = (6a+1)(6b+1) \\ 6x + 7 = (6a-1)(6b-1) \\ 6x + 5 = (6a-1)(6b+1)\end{cases} $$ In other words, either $6x+5$ or $6x+7$ should have a non-trivial factorization, i.e., should not be a prime.
As such, your question is equivalent to the twin prime conjecture, which currently stands open.